Synthetic Division Calculator
Divide a polynomial by x minus k from a coefficient list.
How the synthetic division calculator works
Synthetic division is a compact coefficient-only method for dividing a polynomial by a linear divisor of the form x minus k. The first coefficient drops into the quotient, each new quotient value is multiplied by k, and that product is added to the next coefficient. The final accumulated value is the remainder, which also equals the original polynomial evaluated at k by the remainder theorem.
The default coefficients 1, minus 6, 11, minus 6 represent x cubed minus 6x squared plus 11x minus 6, and k equals 1. Synthetic division produces quotient coefficients 1, minus 5, and 6 with remainder 0. The quotient is therefore x squared minus 5x plus 6, while the zero remainder confirms that x minus 1 is a factor.
Reading synthetic division calculator results privately
Include zero placeholders for missing powers, such as a zero x-squared coefficient in a sparse cubic. This method accepts only a divisor whose leading x coefficient is one; divide a more general linear divisor into monic form first. Decimal coefficients work, but exact rational formatting is not attempted. A tiny nonzero remainder can result from ordinary floating-point rounding. This synthetic division calculator calculation runs entirely in your browser, so the numbers you enter never leave your device.
Frequently Asked Questions
Why must I include zero coefficients?
Synthetic division tracks polynomial degree by position. Omitting a zero for a missing power shifts every later coefficient into the wrong column.
What does a remainder of zero mean?
It means k is a root and x minus k divides the polynomial exactly. The quotient coefficients describe the remaining factor.
Is the synthetic division calculator private?
Yes. Its inputs and results stay in your browser. Bushe.co does not upload or store the values used in this calculation.
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